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1

On oscillatory behaviour of third-order half-linear dynamic equations on time scales

Grace Said R., Chhatria Gokula Nanda

Opuscula Mathematica

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2022

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Vol. 42, no. 6

849--865

EN

In this work, we study the oscillation and asymptotic behaviour of third-order nonlinear dynamic equations on time scales. The findings are obtained using an integral criterion as well as a comparison theorem with the oscillatory properties of a first-order dynamic equation. As a consequence, we give conditions which guarantee that all solutions to the aforementioned problem are only oscillatory, different from any other result in the literature. We propose novel oscillation criteria that improve, extend, and simplify existing ones in the literature. The results are associated with a numerical example. We point out that the results are new even for the case T = R or T = Z.

2

Asymptotic behavior of nonoscillatory solutions of higher-order integro-dynamic equations

Bohner M., Grace S., Sultana N.

Opuscula Mathematica

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2014

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Vol. 34, no. 1

5--14

EN

In this paper, we establish some new criteria on the asymptotic behavior of nonoscillatory solutions of higher-order integro-dynamic equations on time scales.

3

Równanie dynamiczne ruchu kulistego ciała sztywnego w układzie parasola

Polka A

Mechanik

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2012

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R. 85, nr 7CD

815--820

PL

W pracy opisano wzajemne położenie płaszczyzny parasola i płaszczyzny krętu w ruchu kulistym oraz wynikające z tego przyrosty wektorów krętu. Wyprowadzono dynamiczne równanie ruchu kulistego ciała sztywnego w postaci wektorowej dla osi związanych z układem parasola. Wykorzystano zapis w postaci multiiloczynów wektorów oraz macierzowy, z użyciem diad iloczynów skalarnych i wektorowych.

EN

This paper describes relative positions of the umbrella plane and the angular momentum plane of a body in spherical motion as well as the corresponding increments of angular momentum vectors. It derives a dynamic vector equation of spherical motion of a rigid body for the axes bound to the umbrella system. Two notations are used: multiproducts of vectors and matrices (using dyads of scalar and vector products).

4

Numerical Analysis of Open Channel Steady Gradually Varied Flow using the Simplified Saint-Venant Equations

Artichowicz W.

TASK Quarterly : scientific bulletin of Academic Computer Centre in Gdansk

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2011

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Vol. 15, No 3-4

317-328

EN

For one-dimensional open-channel flow modeling, the energy equation is usually used. There exist numerous approaches using the energy equation for open-channel flow computations, which resulted in the development of several very efficient methods for solving this problem applied to channel networks. However, the dynamic equation can be used for this purpose as well. This paper introduces a method for solving a system of non-linear equations by the discretization of the one-dimensional dynamic equation for open-channel networks. The results of the computations using the dynamic and energy equations were compared for an arbitrarily chosen problem. Also, the reasons for the differences between the solution of the dynamic and energy equation were investigated.

5

The asymptotic properties of the dynamic equation with a delayed argument

Cermak J., Urbanek M.

Opuscula Mathematica

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2006

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Vol. 26, no. 3

421-429

EN

In this paper, we present some asymptotic results related to the scalar dynamic equation with a delayed argument. Using the time scale calculus we generalize some results known in the differential and difference case to the more general dynamie case.